Global Spherically Symmetric Classical Solution to Compressible Navier–Stokes Equations with Large Initial Data and Vacuum

Abstract

In this paper, we obtain a result on the existence and uniqueness of global spherically symmetric classical solutions to the compressible isentropic Navier–Stokes equations with vacuum in a bounded domain or exterior domain $\Omega$ of $\mathbb{R}^n$($n\ge2$). Here, the initial data could be large. Besides, the regularities of the solutions are better than those obtained in [H.J. Choe and H. Kim, Math. Methods Appl. Sci., 28 (2005), pp. 1–28; Y. Cho and H. Kim, Manuscripta Math., 120 (2006), pp. 91–129; S.J. Ding, H.Y. Wen, and C.J. Zhu, J. Differential Equations, 251 (2011), pp. 1696–1725]. The analysis is based on some new mathematical techniques and some new useful energy estimates. This is an extension of the work of Choe and Kim, Cho and Kim, and Ding, Wen, and Zhu, where the global radially symmetric strong solutions, the local classical solutions in three dimensions, and the global classical solutions in one dimension were obtained, respectively. This paper can be viewed as the first result on the existence of global classical solutions with large initial data and vacuum in higher dimension.